Analytic functions in the unit ball of bounded value L-distribution in a direction |
|
| Author |
andriykopanytsia@gmail.com
Ivano-Frankivsk National Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine
|
| Abstract |
We present a generalization of concept of bounded value $L$-distribution in direction
$\mathbf{b}\in\mathbb{C}^n\setminus\{0\}$ for
analytic functions in the unit ball $\mathbb{B}^n\subset\mathbb{C}^n$, where $L\colon \mathbb{B}^n \to \mathbb{R}_+$ is a continuous function.
It is established a connection between the class of analytic functions in the unit ball of bounded $L$-index
in the direction $\mathbf{b}$ and the class of analytic functions in the unit ball of bounded
value $L$-distribution in the same direction. The main result of the paper is the following: An analytic function $F\colon \mathbb{B}^n\to\mathbb{C}$ is a function of bounded value $L$-distribution in a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{0\}$ if and only if its directional derivative $\frac{\partial F}{\partial\mathbf{b}}$ has bounded $L$-index in the same direction $\mathbf{b}$.
|
| Keywords |
analytic function; unit ball; bounded index; bounded value distribution
|
| DOI |
doi:10.15330/ms.49.1.75-79
|
| Reference |
1. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire
function of bounded L-index along the direction, Ukrain. Mat. J., 69 (2017), ¹3, 500–508.
2. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci., 227 (2017), ¹1, 1–12. 3. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I.E. Chyzhykov, 2016, 128 p. 4. A.I. Bandura, The properties of entire functions of bounded value L-distribution in direction Prykarp. Visn. Nauk. Tov. Im. Shevchenka, 1(13) (2011), 50-55. 5. A. Bandura, O. Skaskiv, Analytic in the unit ball functions of bounded L-index in direction, https://arxiv.org/abs/1501.04166 6. G.H. Fricke, A note on bounded index and bounded value distribution, Indian J. Pure Appl. Math., 11 (1980), ¹4, 428-432. 7. G.H. Fricke, S.M. Shah, On bounded value distribution and bounded index, Nonlinear Anal., 2 (1978) ¹4, 423-435. 8. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973) ¹1, 117-137. 9. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), ¹1, 3-8. 10. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island, 2 (1968), 298-307. 11. R. Roy, S.M. Shah, The product of two functions of bounded value distribution, Indian J. Pure Appl. Math., 17 (1986), ¹5, 690-693. 12. R. Roy, S.M. Shah, Functions of bounded index, bounded value distribution and v-bounded index, Nonlinear Anal., 11 (1987), 1383-1390. 13. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p. 14. S. Shah, Entire functions of bounded value distribution and gap power series In: P. Erd.os, L. Alpar, G. Halasz, A. Sarkozy (eds.) Studies in Pure Mathematics To the Memory of Paul Turan, p. 629-634, Birkhauser, Basel, 1983. 15. M.N. Sheremeta, An l-index and an l-distribution of the values of entire functions, Soviet Math. (Iz. VUZ), 34 (1990), ¹2, 115-117. |
| Pages |
75-79
|
| Volume |
49
|
| Issue |
1
|
| Year |
2018
|
| Journal |
Matematychni Studii
|
| Full text of paper | |
| Table of content of issue |